Multiple-Scale Analysis and Renormalization of Quenched Second Order Phase Transitions

A quenched second order phase transition is modeled by an effective $\Phi^4$-theory with a time-dependent Hamiltonian $\hat{H} (t)$, whose symmetry is broken spontaneously in time. The quantum field evolves out of equilibrium (nonequilibrium) during the phase transition as the density operator significantly deviates from $\hat{\rho} (t) = e^{- \beta \hat{H} (t)}/Z_H$. The recently developed Liouville-von Neumann (LvN) method provides various quantum states for the phase transition in terms of a complex solution to the mean-field equation, which is equivalent to the Gaussian effective potential in the static case and to the time-dependent Hartree-Fock equation in the nonequilibrium case. Using the multiple scale perturbation theory (MSPT) we solve analytically the mean-field equation to the first order of coupling constant and find the quantum states during the quenched second order phase transition. We propose a renormalization scheme during the process of phase transition to regularize the divergences, which originate from the mode coupling between hard and hard modes or between the soft and hard modes. The effect of mode coupling is discussed.